Fig 1 Multichannel signal transport performance in the Hofstadter model.
Published:31 October 2024,
Published Online:02 September 2024,
Received:04 February 2024,
Revised:16 July 2024,
Accepted:25 July 2024
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The presence of long-range interactions is crucial in distinguishing between abstract complex networks and wave systems. In photonics, because electromagnetic interactions between optical elements generally decay rapidly with spatial distance, most wave phenomena are modeled with neighboring interactions, which account for only a small part of conceptually possible networks. Here, we explore the impact of substantial long-range interactions in topological photonics. We demonstrate that a crystalline structure, characterized by long-range interactions in the absence of neighboring ones, can be interpreted as an overlapped lattice. This overlap model facilitates the realization of higher values of topological invariants while maintaining bandgap width in photonic topological insulators. This breaking of topology-bandgap tradeoff enables topologically protected multichannel signal processing with broad bandwidths. Under practically accessible system parameters, the result paves the way to the extension of topological physics to network science.
Nontrivial topological states of electrons
A breakthrough could be found in a previously overlooked degree of freedom in topological systems—long-range connectivity, especially, the systems possessing stronger long-range interactions than short-range ones. In network science, it was recently proved that a Laplacian corresponding to an arbitrary fully-connected network can provide a free-form level statistics, demonstrating the inverse design of bandgaps with designed gap widths
Here, we exploit substantial long-range interactions to break channel-bandwidth limit in topological photonics. As an extreme scenario, we focus on long-range connectivity in the absence of nearest-neighbor interactions. To analyze such long-range connectivity, we propose the lattice overlap model: the overlap of unit lattices having nearest-neighbor interactions. This lattice overlap strategy provides the design freedom for achieving arbitrary Chern number while preserving the bandgap width, as demonstrated in the Hofstadter model example. Using the system parameters available in conventional silicon photonics, we demonstrate the incoherent optical functionality—designed manipulation of multichannel light with random phases and amplitudes—which is robust to various types of disorder originating from manufacturing defects. Achieving noise-immune signal processing with enhanced information capacity, our design principle paves the way to implementing topological phenomena in complex networks.
Before examining long-range connectivity, we revisit the Hofstadter model
1
Fig 1 Multichannel signal transport performance in the Hofstadter model.
a A schematic of a coupled-resonator square lattice with nearest-neighbor interactions between pseudospin modes (σ = +1). b The Hofstadter butterfly, representing the gap Chern numbers C with different colors. An example of the wing segment for C = 3 is illustrated with the red outlines. The color bar represents C. c The tradeoff between C and AC,i. The red and blue lines denote the C-AC,i relationships for the largest and smallest wings, respectively. The gray dots depict the areas of intermediate-sized wings
where ω0 is the resonance frequency of each resonator,
2
where (m1, m2) and (n1, n2) are the integer indices denoting the positions of the mth and nth sites, respectively, and δa,b is the Kronecker delta function. The designed hopping phase can be implemented with non-resonant waveguide loops
The topological invariant determining the number of topologically protected edge modes
3
where Cl(p,q) is well-defined except for gap closing with even q and l = q/2. The colored Hofstadter butterfly
According to Eq. (3), Cl can have an arbitrary integer by controlling the flux parameters p and q, and the bandgap number l, allowing any numbers of topologically protected edge modes within the frequency range of ω0 – 4t < ω < ω0 + 4t. However, increasing the number of edge modes substantially decreases the bandgap width (Δω in
To quantify the tradeoff relation, first we investigate the number of butterfly wings—or topological bandgaps—NC for a specific gap Chern number C ≡ Cl: NC = φ(1) + ··· + φ(2 | C |) for the Euler's phi function φ(n) (Supplementary Note S3). The performance of C-channel signal transport using topologically protected edge modes can then be characterized by the area AC,i of the ith wing (i = 1, 2, …, NC) for each C, where
4
Δω(α) is the bandgap width at α, and αC,iU and αC,iL are the upper and lower bounds of the wing, respectively, satisfying the gap closing at Δω(αC,iU) = Δω(αC,iL) = 0. We numerically calculate AC,i for C ≤ 15 by approximating each wing as a 60-gon (Supplementary Note S3).
To overcome the limit of the signal transport performance AC,i, we exploit long-range connectivity preserving discrete translational symmetry. We focus on a class of the long-range connectivity that can be decomposed into the decoupled overlap of the translated lattices having only nearest-neighbor interactions. For example, overlapping the identical lattices with in-plane translation allows for the coexistence of their eigenstates. Because Δω and Δα maintain their values of an individual lattice due to decoupling, the increase of the gap Chern number C is expected while preserving the performance figure AC,i. Importantly, the vertical stacking of identical two-dimensional (2D) systems, which has recently become a substantial idea for achieving high Chern numbers in magnetic topological insulator films
To employ the overlap of lattices, we apply the following two criteria for real implementation. First, the sites and connections of the lattices should not be overlapped to maintain the band properties of the original lattices. Second, the pointwise crossings between the connections are allowed by employing the waveguide crossing structures with negligible crosstalks and losses. We demonstrate our proposal by overlapping two lattices as an example.
Fig 2 Overlapped Hofstadter lattices for long-range connectivity.
a Schematics of the two decoupled square lattices having nearest-neighbor interactions (left) and the overlapped lattice having only next-nearest-neighbor interactions (right). Orange circles and gray lines denote the site resonators and non-resonant waveguide couplers. The lattice constant of each composing lattice is d. The black box on the overlapped lattice indicates the crossing between long-range interactions. b The silicon waveguide implementation
The Hamiltonian of the overlapped lattice is given by the direct sum as:
5
where H1 and H2 are the Hamiltonians of the Hofstadter model represented by Eq. (1), corresponding to the top and bottom square lattices in
The hurdle in realizing the overlapped lattice is the crossings between long-range interactions (black box in
To demonstrate our proposal, we calculate the maximum wing area AC,1 for different numbers of the overlapped lattices (
To explore nontrivial topological natures of overlapped lattices, we investigate the edge modes at lattice boundaries. We compare three different configurations: the surface of a bulk (
Fig 3 Edge mode analysis.
a Field profile of the topologically nontrivial edge mode in the overlapped lattice. The domain has α = 1/3 for each sublattice, leading to C = 2 in the lower gap. The black arrow indicates the propagating direction of the mode. b Field profile of the topologically nontrivial edge mode at the interface between two overlapped lattices. The adjacent domains have inhomogeneous Chern numbers (Cbottom = 2 and Ctop = −1). a, b denote the star markers in (d, e), respectively. c The absence of the topologically nontrivial edge mode at the interface between two overlapped lattices with homogeneous Chern numbers (Cbottom = Ctop = 1). d−f The band structures of the systems in (a−c), respectively. kx denotes the x-axis component of the reciprocal vector in the magnetic Brillouin zone. ymean denotes the y-axis component of the center-of-mass of the corresponding modes. The dashed lines in (d) represent the ensemble averaged dispersions under 500 realizations of uniformly random diagonal disorder [–Udiag, +Udiag] on the resonance frequency, where Udiag = 0.5t. All the results are obtained with the tight-binding equation with the supercell technique (Supplementary Note S6)
Edge mode dispersions in
To confirm the enhanced information capacity in signal transport through overlapped topological lattices, we analyze the scattering of edge modes. We examine the system that includes three interfaces between the overlapped lattices, thereby composing an edge-mode beam splitter (
Fig 4 Incoherent multichannel beam splitting.
a A schematic of the scattering region. Three domains possess α = ±1/3 to support the gap Chern number ±1 at the shared bandgap range. A 101 × 101 scattering region is assumed for the analysis in (c−e). b The scattering region with semi-infinite ports for applying the scattering matrix method. The orange arrows represent the centered edge modes of each port. The black arrows denote the outer edge modes resulting from finite widths of the ports. The incoherent transmittance is proportional to the number of centered edge modes. c Intensity profile for an incoherent light at the frequency ω = ω0 − 1.5t. The input is obtained with 100 random linear combinations of the input edge modes. d Intensity profile against diagonal disorder. The resonance frequency of each site in the scattering region is perturbed uniformly between [−Udiag, +Udiag]. e The transmittances for the incoherent incidence against diagonal disorder. The transmittance curves for 50 realizations of diagonal disorder are overlaid. Udiag = 1.5t in (d, e). The shaded region in (e) denotes the frequency range of perfect beam splitting operation against disorder
To fully exploit the superior performance of the overlapped lattices—large AC,i with broader bandgaps and multiple edge modes—we focus on topologically protected wave functionalities applied to spatio-temporally incoherent light sources. The light source is defined by the superposition of the edge modes having uniformly random amplitudes and phases. To quantify the energy flow throughout the kth port (k = 1, 2, 3), we define incoherent transmittance Tk as the average of the squares of the scattering coefficients between port 1 and port k:
6
where Nin,1 is the number of the input modes at port 1, Nout,k is the number of the output modes throughout port k. Because the centered edge modes (orange arrows in
From the unitary sub-scattering matrix, Tk in the frequency region supporting the centered edge modes is solely determined by the number of centered edge modes under the incoherent incidences (
We have exploited the long-range connectivity to break channel-bandwidth limit in topological photonics. By quantitatively analyzing the Hofstadter butterfly, we have revealed the power laws between the gap Chern numbers and the area of the wings. The observed quadratic scaling underlies the necessity of breaking the tradeoff between the bandgap and Chern number. We have modeled the long-range interactions as the overlap of the lattices having nearest-neighbor interactions, which enables the breaking of the tradeoff as demonstrated in the band analysis. Employing this feature, we have demonstrated the defect-robust beam splitting functionality applicable to incoherent light. All the system parameters are devised within the regime of conventional silicon photonics technology.
In evaluating the performance of our overlapped lattice model, it is necessary to examine the relationship among device bandwidth and lattice sizes (Supplementary Note S9). Notably, the spectral bandwidth of the Hofstadter model is primarily governed by the hopping constant t, which determines the bandgap width (Supplementary Fig. S14). However, the variation of lattice sizes due to the lattice overlap eventually affects the device bandwidth. As illustrated in Supplementary Fig. S15a, our overlap model increases the distance between the connected resonators even under an ideal condition—namely, neglecting the spatial footprint of waveguide crossing structures. Furthermore, to guarantee sufficiently large spectral bandwidths during the hopping process, we employ an adiabatic design for waveguide crossing structures with slowly-varying shapes (
Besides the proposed silicon-waveguide implementation, alternative platforms are available for realizing the overlapped lattices. First, optical resonators under spatio-temporal modulations can be applied to construct the overlapped lattices across both spatial and synthetic dimensions
The generality of lattice overlap modeling also inspires further studies. First, the overlap is applicable to other topological systems with different symmetries such as the Haldane
We implement the overlapped lattice composed of ring resonators, which are pair-wisely coupled via off-resonant waveguide loops. A pair of coupled ring resonators is described by the following tight-binding Hamiltonian
7
where a†, a, b†, and b are the creation and annihilation operators of the resonator 'a' and 'b', ω0 is the resonance frequency, τ is the energy leakage rate from each resonator to the evanescently coupled waveguide, and φ is the tunable phase shift obtained through the waveguide loop (Supplementary Note S1 for derivation from the temporal coupled mode theory).
We color the Hofstadter butterflies in
We perform the FDTD full-wave analysis using Tidy3D
To extract the scattering matrix between the edge modes at the center junction in
We acknowledge financial support from the National Research Foundation of Korea (NRF) through the Basic Research Laboratory (No. RS-2024-00397664), Young Researcher Program (No. 2021R1C1C1005031) and Mid-career Researcher Program (No. RS-2023-00274348), all funded by the Korean government. This work was supported by Creative-Pioneering Researchers Program and the BK21 FOUR program of the Education and Research Program for Future ICT Pioneers in 2023, through Seoul National University. We also acknowledge an administrative support from SOFT foundry institute.
G.K. and S.Y. conceived the idea. G.K. developed the theoretical tool and performed the numerical analysis. J.S. and D.L. examined the theoretical and numerical analysis. S.Y. and N.P. supervised the findings of this work. All authors discussed the results and wrote the final manuscript.
The data that support the plots and other findings of this study are available from the corresponding author upon request.
All code developed in this work will be made available from the corresponding author upon request.
The authors declare no competing interests.
Supplementary information The online version contains supplementary material available at https://doi.org/10.1038/s41377-024-01557-4.
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